12.2 Limiting efficiency of transmission systems and Shannon bound
Indicators
and
make sense a specific rates,
and inverse values =1/
and =1/
define specific expenses
of corresponding resources on an information transferring with unity
rate (1 bit per second).
For the Gaussian channel with
frequency band Fch,
the ratio of signal to noise =Ps/Pn
and channel capacity
it
is possible to establish, that these efficiency
indicators are connected by the relation:
and
(12.4)
For ideal system (=1) limiting equation can be defined. According to Shannon theorem by the corresponding transmission methods (coding and modulation) and receiving (demodulation and decoding), the value can be as much as close to unit. Thus the error can be made as much as small. In this case by a condition = 1 it is received limiting equation between and :
. (12.5)
This formula defines of energy efficiency from the frequency efficiency for the ideal system ensuring equality of a information rate to a channel capacity. It is convenient to represent this equation in the form of a curve on a plane = f() (figure 12.1, a curve «Shannon bound»). This curve is limiting and reflects the best interchanging between and in the continuous channel (СC).
It is necessary to notice, that frequency efficiency varies in limits from 0 to , energy efficiency is bounded above by magnitude:
. (12.6)
Differently, energy efficiency of any information transmitting system in a Gaussian channel can not exceed the magnitude
. (12.7)
Similar limiting curves can be constructed and for any other channels if in formulas (12.2) and (12.3) instead of a rate Rchan to substitute expressions for a channel capacity of the corresponding channel. So, in particular, on fig. 12.1 the curve for limiting equation =f () the is discrete-continuous channel (D-CC) is shown.
It "is enclosed" in a curve of the continuous channel (CC) that confirms know result of an information theory according to which DN channel capacity of D-CC always is less a than channel capacity of the continuous channel (CC) which is a basis for construction of corresponding D-CC. In real digital systems error probability p always has a final value and informational efficiency is less then a limiting value max. In these cases for the fixed error probability p = const it is possible to define efficiency ratio to and to construct curves = f ().
In coordinates (, ) to each variant of a transmission system there will corresponds a point on a plane. All these points (curves) should place below a limiting curve of «Shannon bound». The place of these curves depends on an aspect of signals (modulations), a codes (coding methods) and a method of the elaborating of a signals (demodulation/decoding). About perfection of the digital telecommunication methods judge on a degree of placing of real efficiency of to the limiting values.
Concrete data about the efficiency of various modulation/coding methods and also their combinations are given in following section.